In this paper the ray-gridding approach, a new numerical technique for the stability analysis of linear switched systems is presented. It is based on uniform partitions of the state-space in terms of ray directions which allow refinable families of polytopes of adjustable complexity to be examined for invariance. In this framework the existence of a polyhedral Lyapunov function that is common to a family of asymptotically stable subsystems can be checked efficiently via simple iterative algorithms. The technique can be used to prove the stability of switched linear systems, classes of linear time-varying systems and linear differential Inclusions. We also present preliminary results on another related problem; namely, the construction of multiple polyhedral Lyapunov functions for the specification of stabilizing sit,itching sequences for a switched system constructed from a family of stable linear subsystems.